I have seen in a couple places people are using Kd = 2*sqrt(Kp). Kd is derivative gain and Kp is proportional gain (in the context of a PD controller). Does anyone know where this comes from or can provide some intuition for why this is used?
Thanks in advance
Sometimes in introductory classes for PID, the double integrator plant,
$$G(s) = \frac{1}{s^2}$$
is introduced as an example to control. It models a physical system with no external forces but an applied force. It can be used to provide intuition behind the effects of PID control. With this specific plant and ideal PD control law,
$$C(s) = K_d\,s + K_p,$$
placed in negative feedback, we have the closed-loop transfer function from reference to output is,
$$\frac{K_d\,s + K_p}{s^2 + K_d\,s + K_p}.$$
The closed loop system is stable for all positive gains. The choice $K_d = 2\,\sqrt{K_p}$ ensures you have critically damped behaviour. It should not be used as a general rule since it only applies to double integrator systems. For instance, suppose the plant was instead,
$$G(s) = \frac{1}{s\,(s + b)},$$
where $b > 0$ so that the plant is stable. This can model a physical system with internal damping. Then in closed-loop with the ideal PD law,
$$\frac{K_d\,s + K_p}{s^2 + (K_d + b)\,s + K_p}.$$
If you pick $K_d = 2\,\sqrt{K_p},$ you will end up with an overdamped response. It is not optimal, but not going to add oscillations, and for this reason such a choice does not cause harm but is rarely good.
This discussion is still restricted to two-pole plants without zeros, so you can see how restricted such a rule is in applicability.